# Polynomial Explorations

### updated March 2014

0. Introduction 1 Lagrange Polynomials 2 Chord-Based Constructions 3 Cobwebs 4 Mamikon's Theorem
Under initial construction in preparation for BCME 2014

Full BCME 2014 Applets

Note: to run the applets you need a java-enabled browser (Safari works for me)

## 0. Introduction

These pages offer a collection of explorations into properties of polynomials, starting from the intuitive notion of a chord, and exploring constructions based on chords.

One of the key ideas is that of multiple ways of thinking about the same idea. For example,
• a graph can be perceived as a geometrical object in its own right, and as a collection of points, namely {[x, f(x)], x in some interval of R}.
Developing flexibility to move back and forth between these perceptions is necessary in order to appreciate functions in general and the calculus in particular.
• the collection of all chords of a polynomial can be thought of either as a collection of families of chords, each family consisting of all the chords with a fixed end, or each family consisting of all chords of a fixed interval width.
Developing flexibility to move back and forth between these perceptions contributes to a rich appreciation and comprehension of concepts in the calculus and of functions in general.

Where possible each section is initiated by an animation. Studying the animation, saying to yourself or to others what you see happening (Say What You See), and locating what is changing and what relationships seem to remain invariant, provides conjectures as starting points for exploration. Applets are available to support exploration in directions that I have thought of, though for maximal learning potential, constructing your own applets brings you into contact with important details and so enriches your appreciation, comprehension and understanding.

Some initiating diagrams and animations

 Predict Rational Polynomial Cubic Tangents Tangent Power Chordal Midpoints The red curve is to be the denominator; the green curve is to be the numerator Predict the graph of their quotient (worksheet) The yellow point is the midpoint of two of the roots. Link to Animation For a given point P, how many tangents to f go through point P? Describe the regions of constant Tangent Power. Animation Download Animation What is the locus of midpoints of all chords of a cubic? Animation for Quadratic DownLoad Animation See Lagrange Polynomials 1.3 See Chord-Based Constructions 2.2 See Chord-Based Constructions 2.2

## 1. Lagrange Polynomials

A collection of explorations introducing Lagrange polynomials (the polynomial of least degree through a given set of points)
1.1 Predict the graph
1.2 Lagrange Polynomials and Polynomial Coefficients
1.3 Rational Polynomials
1.4 Actions on Graphs (translating, rotating, scaling)

## 2. Chord-Based Constructions

A collection of explorations using constructions on chords:
2.1 Slope and angle
2.2 Parallel Chords & Tangents
2.3 Chordal Gaps
2.4 Chordal Areas
2.5 Chordal SubTangents & SubNormals
2.6 Chordal Circles
2.7 Chordal Parabolae

## 3. Cobwebs

A collection of explortions of cobwebs involving one or more functions
3.1 Cobwebs on two functions
3.2 Iteration
3.3 Conjugation

## 4. Mamikon's Theorem

Explorations of Mamikon's theorem regarding the area swept out by a tangent
4.1 Fixed Length Tangents